Foundations of Mathematical Thinking · 1st Class

Active learning ideas

Addition and Subtraction as Opposites

Active learning helps students see the reversible nature of addition and subtraction because they physically move objects and record their actions. When children manipulate counters, number lines, or balance scales, they build mental images of how adding and subtracting undo each other. These concrete experiences prevent abstract confusion later.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Algebra - A.2.1
• NCCA: Junior Cycle - Strand 3: Algebra - A.2.2

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Concept Mapping
• 30 min · Pairs

Partner Counters: Add and Undo

Pairs use 20 counters. One student makes a pile of 8, adds 5 more, then subtracts 5 to check return to 8. Partners swap roles and record fact families. Discuss patterns as a class.

What happens when you add a number and then subtract the same number?

Facilitation Tip

During Partner Counters, ask each pair to take turns saying the full fact family aloud after every move to reinforce the connection between the three numbers.

What to look for

Provide students with a number sentence, for example, 7 + 4 = 11. Ask them to write one related subtraction sentence using the same three numbers and explain in one sentence how they are related.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 02

• Concept Mapping
• 25 min · Whole Class

Whole Class Number Line Relay

Mark a floor number line to 20. Call an addition fact like 4 + 7. Students hop forward from 0, note 11, then hop back 7 steps. Repeat with subtraction starts.

How does knowing 3 + 5 = 8 help you work out 8 − 5?

Facilitation Tip

For the Number Line Relay, have the whole class call out the equation together after each jump to keep everyone engaged and accountable.

What to look for

Present students with a simple equation like 15 - 6 = __. Ask them to solve it and then write the corresponding addition equation that proves their answer. Observe their process.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 03

• Concept Mapping
• 35 min · Small Groups

Small Groups: Fact Family Cards

Groups draw three numbers like 9, 4, 13 on cards. Match to make two addition and two subtraction sentences. Sort into family houses and share one with class.

Can you write an addition fact and a subtraction fact using the same three numbers?

Facilitation Tip

When using Fact Family Cards, circulate and listen for students who group the same three numbers without prompting, showing they see the relationship.

What to look for

Ask students: 'If you have 9 counters, add 3 more, and then take away those same 3 counters, how many do you have? Why does this happen?' Listen for explanations of operations undoing each other.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 04

• Concept Mapping
• 20 min · Individual

Individual Balance Scales

Each student uses linking cubes on a balance scale. Add 3 to one side of 10, then remove 3 to balance again. Write the pair of facts and check with a peer.

What happens when you add a number and then subtract the same number?

What to look for

Provide students with a number sentence, for example, 7 + 4 = 11. Ask them to write one related subtraction sentence using the same three numbers and explain in one sentence how they are related.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

A few notes on teaching this unit

Teachers should model the language of undoing each time students add or subtract, using precise phrases like ‘we added 6 and now we subtract 6 to return to 7.’ Avoid using the terms ‘turn-around facts’ or ‘flip facts’ because they blur the idea of reversibility. Research shows that when students physically undo an action, their brains form stronger connections between the operations, so resist rushing to abstract symbols before they are ready.

By the end of these activities, students should confidently state that adding then subtracting the same number returns to the start, write matching fact families, and explain why 8 - 5 can be solved by knowing 5 + 3 = 8. They should also use their hands-on work to justify their answers during discussions.

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Watch Out for These Misconceptions

• During Partner Counters, watch for students who focus only on the final count and miss the step-by-step undoing of addition.

  Prompt them to retrace their moves aloud: ‘You started with 7, added 6 to get 13, and now you subtract 6 to return to 7. Say each step while you move your counters.’

• During Whole Class Number Line Relay, watch for students who step backward the same distance they stepped forward regardless of direction.

  Pause the relay and ask them to trace the path with their finger while saying, ‘I went forward 4 steps from 7 to 11, so to undo I must go backward 4 steps from 11 to 7.’

• During Fact Family Cards, watch for students who group numbers without connecting the operations.

  Have them place one addition card and one subtraction card side by side and read them together, pointing to the numbers as they say, ‘This shows 5 + 3 = 8 and 8 - 3 = 5.’

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Methods used in this brief

• Concept Mapping